Olympiad problems

2010 AMC 12/AHSME, 19

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

2006 AIME Problems, 11

Tags: 3d geometry , ratio , modular arithmetic , geometry , induction

A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: $\bullet$ Any cube may be the bottom cube in the tower. $\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?

2024 Abelkonkurransen Finale, 1b

Tags: residue , function , modular arithmetic , number theory

Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that the numbers \[n, f(n),f(f(n)),\dots,f^{m-1}(n)\] are distinct modulo $m$ for all integers $n,m$ with $m>1$. (Here $f^k$ is defined by $f^0(n)=n$ and $f^{k+1}(n)=f(f^{k}(n))$ for $k \ge 0$.)

1973 Miklós Schweitzer, 1

Tags: abstract algebra , modular arithmetic , galois theory , superior algebra , matrix , group theory , linear algebra

We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$. [i]J. Erdos[/i]

2001 National Olympiad First Round, 3

Tags: quadratic , modular arithmetic

How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the preceding} $

1995 IMO Shortlist, 2

Tags: number theory , quadratic , partition , modular arithmetic

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

1993 Korea - Final Round, 3

Tags: modular arithmetic , number theory

Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.

2010 Iran MO (3rd Round), 1

Tags: modular arithmetic , number theory

suppose that $a=3^{100}$ and $b=5454$. how many $z$s in $[1,3^{99})$ exist such that for every $c$ that $gcd(c,3)=1$, two equations $x^z\equiv c$ and $x^b\equiv c$ (mod $a$) have the same number of answers?($\frac{100}{6}$ points)

2013 Korea National Olympiad, 3

Tags: calculus , polynomial , quadratic , modular arithmetic , algebra , integration , induction

Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that (1) For all integer $m$, $f(m) \ne 0$. (2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.

2010 National Olympiad First Round, 30

Tags: floor function , modular arithmetic

If $N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor +\dots \lfloor \frac{2^{2009}}{5} \rfloor$, what is the remainder when $2^{2010}$ is divided by $N$? $ \textbf{(A)}\ 5034 \qquad\textbf{(B)}\ 5032 \qquad\textbf{(C)}\ 5031 \qquad\textbf{(D)}\ 5028 \qquad\textbf{(E)}\ 5024 $

2010 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: modular arithmetic , number theory

Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.

1998 Romania Team Selection Test, 1

Tags: modular arithmetic , ratio , parallelogram , trapezoid , geometry , incenter , trigonometry

We are given an isosceles triangle $ABC$ such that $BC=a$ and $AB=BC=b$. The variable points $M\in (AC)$ and $N\in (AB)$ satisfy $a^2\cdot AM \cdot AN = b^2 \cdot BN \cdot CM$. The straight lines $BM$ and $CN$ intersect in $P$. Find the locus of the variable point $P$. [i]Dan Branzei[/i]

1988 IMO Longlists, 75

Tags: combinatorics , modular arithmetic

Let $S$ be an infinite set of integers containing zero, and such that the distances between successive number never exceed a given fixed number. Consider the following procedure: Given a set $X$ of integers we construct a new set consisting of all numbers $x \pm s,$ where $x$ belongs to $X$ and s belongs to $S.$ Starting from $S_0 = \{0\}$ we successively construct sets $S_1, S_2, S_3, \ldots$ using this procedure. Show that after a finite number of steps we do not obtain any new sets, i.e. $S_k = S_{k_0}$ for $k \geq k_0.$

1979 IMO Longlists, 45

Tags: modular arithmetic , function , number theory

For any positive integer $n$, we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$, we have $F(10) = 4$. Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$, but odd if $n$ is divisible by $6$.

2012 Singapore MO Open, 4

Tags: relatively prime , number theory , floor function , modular arithmetic

Let $p$ be an odd prime. Prove that \[1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.\]

2011 Belarus Team Selection Test, 3

Tags: divisibility , modular arithmetic , number theory

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

2003 IMO Shortlist, 7

Tags: recurrence , sequence , divisibility , modular arithmetic , prime , polynomial

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

2009 Tuymaada Olympiad, 1

Tags: algebra , modular arithmetic

A magician asked a spectator to think of a three-digit number $ \overline{abc}$ and then to tell him the sum of numbers $ \overline{acb}$, $ \overline{bac}$, $ \overline{bca}$, $ \overline{cab}$, and $ \overline{cba}$. He claims that when he knows this sum he can determine the original number. Is that so?

2014 Cono Sur Olympiad, 2

Tags: number theory , modular arithmetic

A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].

2017 Iran Team Selection Test, 4

Tags: prime number , modular arithmetic , number theory

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

2006 Purple Comet Problems, 11

Tags: modular arithmetic

Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$.

2012 Online Math Open Problems, 9

Tags: modular arithmetic

At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people? [i]Author: Ray Li[/i]

2010 Germany Team Selection Test, 1

Tags: function , divisibility , modular arithmetic , number theory , algebra

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

2014 Contests, 2

Tags: quadratic residue , quadratic , modular arithmetic , number theory , induction

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

2006 Baltic Way, 16

Tags: modular arithmetic , easy , number theory

Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?